I think the Problem was solved decades ago by D. C. Williams and David Stove, and helped along its way by Michael Tooley a few years ago. I'm not sure why the solution isn't wider-accepted. It depends on a couple of assumptions, but those assumptions are widely accepted elsewhere in life, so it's kind of special pleading to reject them here. Maybe philosophers just like fun problems and aren't as interested in their solutions.

Anyway, it's an objective, mathematical fact that most large samples of some set are mostly representative of that set. (The is the Law of Large Numbers.) Suppose there were an urn with four marbles in it: two red ('R1' and 'R2') and two blue ('B1' and 'B2'). How many two-marble samples can possibly be drawn from it (without replacement)?

R1 R2

R1 B1

R1 B2

R2 B1

R2 B2

B1 B2

Notice that only ##1 and 6 are non-representative of the set. All four of the others are 50-50, just like the set of marbles in the urn itself. Therefore, if you randomly drew two marbles out of the urn, you'd be 2/3 probable to have drawn a representative sample, and only 1/3 probable to have drawn a non-representative sample.

The Law of Large Numbers tells us that this is true a fortiori when you have a very large sample. If you drew four marbles out, your sample is 100% likely to be representative of the set.

This Law is also true even when the total set numbers in the billions, and your sample is only in the thousands. Your sample is still very likely to be similar in proportion to the proportion in the actual set.

Now, in the same way that it would be an unlikely coincidence that you would just happen to draw out a non-representative sample from the urn, the Williams-Stove-Tooley solution says it would be an unlikely coincidence that all the black ravens just happened to be clustered around the last couple thousand years. If we've observed a billion ravens, and there are (say) a trillion ravens that will ever have existed, then the probability that the proportion of black ravens in the gigaraven we've observed is within 3% of the proportion of black ravens in the total teraraven is itself well over 95%.

Therefore, the solution can be summarized like this:

The Law of Large Numbers is true.

If (1) then it's extremely likely that for most of the inductions we care about, we've observed samples that are representative of the total.

If (2) then for most of the inductions we care about, induction is reliable.

Therefore, for most of the inductions we care about, induction is reliable.

In my view, this solution is only available to the rationalist, because it depends on the synthetic a priori proposition that the Law of Large Numbers is sound, i.e. applies to reality.

This is one of the more interesting proposed solutions, but it should be noted that it has sustained numerous very serious objections. And I know you probably don't mean it this way, but this kind of statement:

Maybe philosophers just like fun problems and aren't as interested in their solutions.

Comes across as dismissive and arrogant in the face of those objections. In the same way this does (and yes I know the article is being a bit tongue in cheek). It's easy to claim a solution when you dismiss detractors as not being interested in a solution.

it should be noted that it has sustained numerous very serious objections.

Yes; I endorse it because I think the objections can be answered. I apologize if I implied that there have never been any objections to the solution in question, and if I implied that the solution in question is universally accepted.

Maybe philosophers just like fun problems and aren't as interested in their solutions.

In my experience, most philosophers who know about the Problem of Induction don't know about the Williams-Stove-Tooley solution (nor the Armstrong solution, for that matter), and certainly haven't studied the objections to it. Is your experience different?

You and I may have different views of academic philosophy in general. I think most academic philosophers are irrational about philosophy a lot of the time. Do you disagree?

Of course people can be irrational, and some people have different goals. I do not, however, think the many epistemologists working on the problem of induction over the past 70 years are simply uninterested in solving it. Likely, they find the objections to the view far more convincing- and that deserves mention in a sub like this. Sometimes this consensus 'kills' a view in the popular teaching of philosophy- sometimes for the worse (and yes, among non-specialists the Williams solution is not well-known). Though I'd be far more willing to place the blame of this on a flaw in the argument than a flaw in the vast majority of specialists who worked on this problem for 70 years. I for one think the Williams style objection contains fallacious steps and is too weak to solve the problem Hume was interested in.

It seems to me that no matter how cynical you are about the field, declaring oneself one of the few serious/brilliant/dedicated/level-headed participants and claiming that one's epistemic peers are not serious/brilliant/dedicated/level-headed enough to see the answer will pretty much never be justified. The vast majority of specialists I know take their work (at least in their specialty) very seriously and if anything a somewhat conciliatory position in regards to this massive peer disagreement seems warranted.

Oh, I don't think the epistemologists working on the Problem are uninterested in solving it. And I agree that it's misleading to suggest that there's any kind of consensus that the Williams-Stove-Tooley approach is correct, or even that it's not subject to serious objections. Again, if I said that or implied that, I apologize. The comment to which I replied asked for personal opinions, so that's what I provided.

It seems to me that no matter how cynical you are about the field, declaring oneself one of the few serious/brilliant/dedicated/level-headed participants and claiming that one's epistemic peers are not serious/brilliant/dedicated/level-headed enough to see the answer

I hope I didn't claim anything like that. I think I'm more level-headed or sincere in a certain way about certain topics than lots of philosophers are, but I don't know whether I'm more level-headed and sincere in that way than the majority are about the majority of issues.

My main point was that lots of philosophers (especially those who don't specialize in the philosophy of science or epistemology) don't seem interested in finding a solution to the Problem of Induction. (Do you really want to deny that? Again, poll professionals to see how many know about the Problem of Induction, and then poll them to see how many of them have seriously read about or considered, say, the top three solutions.) There are several philosophical problems that I myself am not very interested in finding solutions to. We can debate about whether those problems are more or less important than the Problem of Induction is.

However, on the subject of cynicism, I would point to the major debates that have lasted for generations and such that there are very smart people on both sides. About these, either

both sides are generally rational, but the problem is too difficult; or

philosophers on at least one side are generally not very rational about that debate.

It seems a priori unlikely that the vast majority of major debates can be described as in (1). We have no a priori reason to expect philosophy per se to be difficult, do we?

Thanks for this. It's interesting you say that this is only available to the rationalist, as the truth of the Law of Large Numbers seems to me, naively, to be justified *a posteriori* or empirically... Can you tell me what I am missing here?

Sure. I think the Law of Large Numbers is justified a priori. It's just knowable a priori that most large samples will be representative. You could perform the thought experiment yourself (although it would rapidly get unwieldy) by extending the 'marble' analogy to more-and-more marbles. You might need a computer to do so, but you could prove that most 1000-fold samples of a set of 10,000 will feature within 10% of the proportion in the 10,000. (These numbers are off the top of my head but you should be able to get more precision from a textbook.)

Now, the Law of Large Numbers might also be justifiable empirically. You could keep drawing marbles out of urns and notice that most of the samples you draw are representative of the total. But how do you know that it will keep being that way? Maybe the next million draws will be unrepresentative. That's why I think you need a priori knowledge here. You need to know a priori that math (I guess ZFC math?) is sound: that it's not just a fun system of (syntactic) rules, but actually maps onto (semantic) reality. And since the system purports to be logically necessary (i.e. e.g. the Law of Large Numbers is a theorem), and we know a priori that ZFC math is sound, we know that necessarily, in real life, most large samples will be representative, and it will stay that way.

I haven't thought very carefully about all the details here, so I may have made a mistake somewhere, but that's how it seems to me.